Properties

Label 637.2.a.j.1.2
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.470683 q^{2} -2.24914 q^{3} -1.77846 q^{4} -0.529317 q^{5} -1.05863 q^{6} -1.77846 q^{8} +2.05863 q^{9} +O(q^{10})\) \(q+0.470683 q^{2} -2.24914 q^{3} -1.77846 q^{4} -0.529317 q^{5} -1.05863 q^{6} -1.77846 q^{8} +2.05863 q^{9} -0.249141 q^{10} -2.24914 q^{11} +4.00000 q^{12} -1.00000 q^{13} +1.19051 q^{15} +2.71982 q^{16} +1.30777 q^{17} +0.968964 q^{18} +1.47068 q^{19} +0.941367 q^{20} -1.05863 q^{22} +5.83709 q^{23} +4.00000 q^{24} -4.71982 q^{25} -0.470683 q^{26} +2.11727 q^{27} +5.22154 q^{29} +0.560352 q^{30} +7.02760 q^{31} +4.83709 q^{32} +5.05863 q^{33} +0.615547 q^{34} -3.66119 q^{36} -2.36641 q^{37} +0.692226 q^{38} +2.24914 q^{39} +0.941367 q^{40} -6.49828 q^{41} +11.3940 q^{43} +4.00000 q^{44} -1.08967 q^{45} +2.74742 q^{46} -8.58451 q^{47} -6.11727 q^{48} -2.22154 q^{50} -2.94137 q^{51} +1.77846 q^{52} +11.2767 q^{53} +0.996562 q^{54} +1.19051 q^{55} -3.30777 q^{57} +2.45769 q^{58} +12.1725 q^{59} -2.11727 q^{60} +2.00000 q^{61} +3.30777 q^{62} -3.16291 q^{64} +0.529317 q^{65} +2.38101 q^{66} -15.9379 q^{67} -2.32582 q^{68} -13.1284 q^{69} +1.19051 q^{71} -3.66119 q^{72} -7.64315 q^{73} -1.11383 q^{74} +10.6155 q^{75} -2.61555 q^{76} +1.05863 q^{78} -1.33881 q^{79} -1.43965 q^{80} -10.9379 q^{81} -3.05863 q^{82} +16.3500 q^{83} -0.692226 q^{85} +5.36297 q^{86} -11.7440 q^{87} +4.00000 q^{88} -6.91033 q^{89} -0.512889 q^{90} -10.3810 q^{92} -15.8061 q^{93} -4.04059 q^{94} -0.778457 q^{95} -10.8793 q^{96} +3.47068 q^{97} -4.63016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9} + 8 q^{10} + 2 q^{11} + 12 q^{12} - 3 q^{13} - 6 q^{15} - q^{16} - 4 q^{17} - 15 q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 10 q^{23} + 12 q^{24} - 5 q^{25} - q^{26} + 8 q^{27} + 24 q^{29} + 20 q^{30} + 4 q^{31} + 7 q^{32} + 16 q^{33} - 14 q^{34} - q^{36} + 10 q^{38} - 2 q^{39} + 2 q^{40} - 2 q^{41} + 10 q^{43} + 12 q^{44} - 22 q^{45} - 18 q^{46} + 8 q^{47} - 20 q^{48} - 15 q^{50} - 8 q^{51} - 3 q^{52} + 8 q^{53} - 32 q^{54} - 6 q^{55} - 2 q^{57} + 12 q^{58} + 4 q^{59} - 8 q^{60} + 6 q^{61} + 2 q^{62} - 17 q^{64} + 2 q^{65} - 12 q^{66} - 12 q^{67} - 22 q^{68} + 6 q^{69} - 6 q^{71} - q^{72} + 10 q^{73} + 30 q^{74} + 16 q^{75} + 8 q^{76} + 4 q^{78} - 14 q^{79} + 14 q^{80} + 3 q^{81} - 10 q^{82} + 12 q^{83} - 10 q^{85} - 26 q^{86} + 26 q^{87} + 12 q^{88} - 2 q^{89} + 28 q^{90} - 12 q^{92} - 22 q^{93} + 10 q^{94} + 6 q^{95} + 4 q^{96} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.470683 0.332823 0.166412 0.986056i \(-0.446782\pi\)
0.166412 + 0.986056i \(0.446782\pi\)
\(3\) −2.24914 −1.29854 −0.649271 0.760557i \(-0.724926\pi\)
−0.649271 + 0.760557i \(0.724926\pi\)
\(4\) −1.77846 −0.889229
\(5\) −0.529317 −0.236718 −0.118359 0.992971i \(-0.537763\pi\)
−0.118359 + 0.992971i \(0.537763\pi\)
\(6\) −1.05863 −0.432185
\(7\) 0 0
\(8\) −1.77846 −0.628780
\(9\) 2.05863 0.686211
\(10\) −0.249141 −0.0787852
\(11\) −2.24914 −0.678141 −0.339071 0.940761i \(-0.610113\pi\)
−0.339071 + 0.940761i \(0.610113\pi\)
\(12\) 4.00000 1.15470
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 1.19051 0.307388
\(16\) 2.71982 0.679956
\(17\) 1.30777 0.317182 0.158591 0.987344i \(-0.449305\pi\)
0.158591 + 0.987344i \(0.449305\pi\)
\(18\) 0.968964 0.228387
\(19\) 1.47068 0.337398 0.168699 0.985668i \(-0.446043\pi\)
0.168699 + 0.985668i \(0.446043\pi\)
\(20\) 0.941367 0.210496
\(21\) 0 0
\(22\) −1.05863 −0.225701
\(23\) 5.83709 1.21712 0.608559 0.793509i \(-0.291748\pi\)
0.608559 + 0.793509i \(0.291748\pi\)
\(24\) 4.00000 0.816497
\(25\) −4.71982 −0.943965
\(26\) −0.470683 −0.0923086
\(27\) 2.11727 0.407468
\(28\) 0 0
\(29\) 5.22154 0.969616 0.484808 0.874621i \(-0.338889\pi\)
0.484808 + 0.874621i \(0.338889\pi\)
\(30\) 0.560352 0.102306
\(31\) 7.02760 1.26219 0.631097 0.775704i \(-0.282605\pi\)
0.631097 + 0.775704i \(0.282605\pi\)
\(32\) 4.83709 0.855085
\(33\) 5.05863 0.880595
\(34\) 0.615547 0.105566
\(35\) 0 0
\(36\) −3.66119 −0.610198
\(37\) −2.36641 −0.389035 −0.194517 0.980899i \(-0.562314\pi\)
−0.194517 + 0.980899i \(0.562314\pi\)
\(38\) 0.692226 0.112294
\(39\) 2.24914 0.360151
\(40\) 0.941367 0.148843
\(41\) −6.49828 −1.01486 −0.507431 0.861693i \(-0.669405\pi\)
−0.507431 + 0.861693i \(0.669405\pi\)
\(42\) 0 0
\(43\) 11.3940 1.73757 0.868785 0.495190i \(-0.164902\pi\)
0.868785 + 0.495190i \(0.164902\pi\)
\(44\) 4.00000 0.603023
\(45\) −1.08967 −0.162438
\(46\) 2.74742 0.405085
\(47\) −8.58451 −1.25218 −0.626090 0.779751i \(-0.715346\pi\)
−0.626090 + 0.779751i \(0.715346\pi\)
\(48\) −6.11727 −0.882951
\(49\) 0 0
\(50\) −2.22154 −0.314174
\(51\) −2.94137 −0.411874
\(52\) 1.77846 0.246628
\(53\) 11.2767 1.54898 0.774490 0.632587i \(-0.218007\pi\)
0.774490 + 0.632587i \(0.218007\pi\)
\(54\) 0.996562 0.135615
\(55\) 1.19051 0.160528
\(56\) 0 0
\(57\) −3.30777 −0.438125
\(58\) 2.45769 0.322711
\(59\) 12.1725 1.58472 0.792360 0.610054i \(-0.208852\pi\)
0.792360 + 0.610054i \(0.208852\pi\)
\(60\) −2.11727 −0.273338
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 3.30777 0.420088
\(63\) 0 0
\(64\) −3.16291 −0.395364
\(65\) 0.529317 0.0656536
\(66\) 2.38101 0.293083
\(67\) −15.9379 −1.94713 −0.973564 0.228415i \(-0.926646\pi\)
−0.973564 + 0.228415i \(0.926646\pi\)
\(68\) −2.32582 −0.282047
\(69\) −13.1284 −1.58048
\(70\) 0 0
\(71\) 1.19051 0.141287 0.0706436 0.997502i \(-0.477495\pi\)
0.0706436 + 0.997502i \(0.477495\pi\)
\(72\) −3.66119 −0.431475
\(73\) −7.64315 −0.894562 −0.447281 0.894393i \(-0.647608\pi\)
−0.447281 + 0.894393i \(0.647608\pi\)
\(74\) −1.11383 −0.129480
\(75\) 10.6155 1.22578
\(76\) −2.61555 −0.300024
\(77\) 0 0
\(78\) 1.05863 0.119867
\(79\) −1.33881 −0.150628 −0.0753139 0.997160i \(-0.523996\pi\)
−0.0753139 + 0.997160i \(0.523996\pi\)
\(80\) −1.43965 −0.160958
\(81\) −10.9379 −1.21533
\(82\) −3.05863 −0.337770
\(83\) 16.3500 1.79464 0.897322 0.441377i \(-0.145510\pi\)
0.897322 + 0.441377i \(0.145510\pi\)
\(84\) 0 0
\(85\) −0.692226 −0.0750825
\(86\) 5.36297 0.578304
\(87\) −11.7440 −1.25909
\(88\) 4.00000 0.426401
\(89\) −6.91033 −0.732494 −0.366247 0.930518i \(-0.619357\pi\)
−0.366247 + 0.930518i \(0.619357\pi\)
\(90\) −0.512889 −0.0540632
\(91\) 0 0
\(92\) −10.3810 −1.08230
\(93\) −15.8061 −1.63901
\(94\) −4.04059 −0.416755
\(95\) −0.778457 −0.0798680
\(96\) −10.8793 −1.11036
\(97\) 3.47068 0.352395 0.176197 0.984355i \(-0.443620\pi\)
0.176197 + 0.984355i \(0.443620\pi\)
\(98\) 0 0
\(99\) −4.63016 −0.465348
\(100\) 8.39400 0.839400
\(101\) −7.75086 −0.771239 −0.385620 0.922658i \(-0.626012\pi\)
−0.385620 + 0.922658i \(0.626012\pi\)
\(102\) −1.38445 −0.137081
\(103\) 16.9966 1.67472 0.837361 0.546651i \(-0.184098\pi\)
0.837361 + 0.546651i \(0.184098\pi\)
\(104\) 1.77846 0.174392
\(105\) 0 0
\(106\) 5.30777 0.515537
\(107\) −5.55691 −0.537207 −0.268604 0.963251i \(-0.586562\pi\)
−0.268604 + 0.963251i \(0.586562\pi\)
\(108\) −3.76547 −0.362332
\(109\) 7.92332 0.758917 0.379458 0.925209i \(-0.376110\pi\)
0.379458 + 0.925209i \(0.376110\pi\)
\(110\) 0.560352 0.0534275
\(111\) 5.32238 0.505178
\(112\) 0 0
\(113\) −9.89229 −0.930588 −0.465294 0.885156i \(-0.654051\pi\)
−0.465294 + 0.885156i \(0.654051\pi\)
\(114\) −1.55691 −0.145818
\(115\) −3.08967 −0.288113
\(116\) −9.28629 −0.862210
\(117\) −2.05863 −0.190321
\(118\) 5.72938 0.527432
\(119\) 0 0
\(120\) −2.11727 −0.193279
\(121\) −5.94137 −0.540124
\(122\) 0.941367 0.0852273
\(123\) 14.6155 1.31784
\(124\) −12.4983 −1.12238
\(125\) 5.14486 0.460171
\(126\) 0 0
\(127\) 0.824101 0.0731271 0.0365635 0.999331i \(-0.488359\pi\)
0.0365635 + 0.999331i \(0.488359\pi\)
\(128\) −11.1629 −0.986671
\(129\) −25.6267 −2.25631
\(130\) 0.249141 0.0218511
\(131\) −10.6155 −0.927485 −0.463742 0.885970i \(-0.653494\pi\)
−0.463742 + 0.885970i \(0.653494\pi\)
\(132\) −8.99656 −0.783050
\(133\) 0 0
\(134\) −7.50172 −0.648050
\(135\) −1.12070 −0.0964549
\(136\) −2.32582 −0.199437
\(137\) 11.3630 0.970804 0.485402 0.874291i \(-0.338673\pi\)
0.485402 + 0.874291i \(0.338673\pi\)
\(138\) −6.17934 −0.526020
\(139\) 13.9233 1.18096 0.590480 0.807052i \(-0.298938\pi\)
0.590480 + 0.807052i \(0.298938\pi\)
\(140\) 0 0
\(141\) 19.3078 1.62601
\(142\) 0.560352 0.0470237
\(143\) 2.24914 0.188083
\(144\) 5.59912 0.466593
\(145\) −2.76385 −0.229525
\(146\) −3.59750 −0.297731
\(147\) 0 0
\(148\) 4.20855 0.345941
\(149\) 9.30777 0.762523 0.381261 0.924467i \(-0.375490\pi\)
0.381261 + 0.924467i \(0.375490\pi\)
\(150\) 4.99656 0.407968
\(151\) 7.07324 0.575612 0.287806 0.957689i \(-0.407074\pi\)
0.287806 + 0.957689i \(0.407074\pi\)
\(152\) −2.61555 −0.212149
\(153\) 2.69223 0.217654
\(154\) 0 0
\(155\) −3.71982 −0.298783
\(156\) −4.00000 −0.320256
\(157\) 6.04059 0.482091 0.241046 0.970514i \(-0.422510\pi\)
0.241046 + 0.970514i \(0.422510\pi\)
\(158\) −0.630155 −0.0501325
\(159\) −25.3630 −2.01141
\(160\) −2.56035 −0.202414
\(161\) 0 0
\(162\) −5.14830 −0.404489
\(163\) 6.38101 0.499800 0.249900 0.968272i \(-0.419602\pi\)
0.249900 + 0.968272i \(0.419602\pi\)
\(164\) 11.5569 0.902443
\(165\) −2.67762 −0.208452
\(166\) 7.69566 0.597299
\(167\) 16.5845 1.28335 0.641674 0.766977i \(-0.278240\pi\)
0.641674 + 0.766977i \(0.278240\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −0.325819 −0.0249892
\(171\) 3.02760 0.231526
\(172\) −20.2637 −1.54510
\(173\) 23.3009 1.77153 0.885767 0.464130i \(-0.153633\pi\)
0.885767 + 0.464130i \(0.153633\pi\)
\(174\) −5.52770 −0.419054
\(175\) 0 0
\(176\) −6.11727 −0.461106
\(177\) −27.3776 −2.05782
\(178\) −3.25258 −0.243791
\(179\) 21.0422 1.57277 0.786384 0.617738i \(-0.211951\pi\)
0.786384 + 0.617738i \(0.211951\pi\)
\(180\) 1.93793 0.144445
\(181\) −16.7474 −1.24483 −0.622413 0.782689i \(-0.713848\pi\)
−0.622413 + 0.782689i \(0.713848\pi\)
\(182\) 0 0
\(183\) −4.49828 −0.332523
\(184\) −10.3810 −0.765299
\(185\) 1.25258 0.0920914
\(186\) −7.43965 −0.545501
\(187\) −2.94137 −0.215094
\(188\) 15.2672 1.11347
\(189\) 0 0
\(190\) −0.366407 −0.0265819
\(191\) −7.43965 −0.538314 −0.269157 0.963096i \(-0.586745\pi\)
−0.269157 + 0.963096i \(0.586745\pi\)
\(192\) 7.11383 0.513396
\(193\) −1.50172 −0.108096 −0.0540480 0.998538i \(-0.517212\pi\)
−0.0540480 + 0.998538i \(0.517212\pi\)
\(194\) 1.63359 0.117285
\(195\) −1.19051 −0.0852540
\(196\) 0 0
\(197\) 23.9931 1.70944 0.854720 0.519090i \(-0.173729\pi\)
0.854720 + 0.519090i \(0.173729\pi\)
\(198\) −2.17934 −0.154879
\(199\) 2.01461 0.142812 0.0714059 0.997447i \(-0.477251\pi\)
0.0714059 + 0.997447i \(0.477251\pi\)
\(200\) 8.39400 0.593546
\(201\) 35.8466 2.52843
\(202\) −3.64820 −0.256687
\(203\) 0 0
\(204\) 5.23109 0.366250
\(205\) 3.43965 0.240235
\(206\) 8.00000 0.557386
\(207\) 12.0164 0.835199
\(208\) −2.71982 −0.188586
\(209\) −3.30777 −0.228803
\(210\) 0 0
\(211\) 10.1008 0.695370 0.347685 0.937611i \(-0.386968\pi\)
0.347685 + 0.937611i \(0.386968\pi\)
\(212\) −20.0552 −1.37740
\(213\) −2.67762 −0.183467
\(214\) −2.61555 −0.178795
\(215\) −6.03104 −0.411313
\(216\) −3.76547 −0.256208
\(217\) 0 0
\(218\) 3.72938 0.252585
\(219\) 17.1905 1.16163
\(220\) −2.11727 −0.142746
\(221\) −1.30777 −0.0879704
\(222\) 2.50516 0.168135
\(223\) −10.1414 −0.679120 −0.339560 0.940584i \(-0.610278\pi\)
−0.339560 + 0.940584i \(0.610278\pi\)
\(224\) 0 0
\(225\) −9.71639 −0.647759
\(226\) −4.65613 −0.309721
\(227\) 5.38445 0.357379 0.178689 0.983906i \(-0.442814\pi\)
0.178689 + 0.983906i \(0.442814\pi\)
\(228\) 5.88273 0.389594
\(229\) −3.32238 −0.219549 −0.109775 0.993957i \(-0.535013\pi\)
−0.109775 + 0.993957i \(0.535013\pi\)
\(230\) −1.45426 −0.0958908
\(231\) 0 0
\(232\) −9.28629 −0.609675
\(233\) 13.7198 0.898816 0.449408 0.893327i \(-0.351635\pi\)
0.449408 + 0.893327i \(0.351635\pi\)
\(234\) −0.968964 −0.0633432
\(235\) 4.54392 0.296413
\(236\) −21.6482 −1.40918
\(237\) 3.01117 0.195597
\(238\) 0 0
\(239\) −3.50172 −0.226507 −0.113254 0.993566i \(-0.536127\pi\)
−0.113254 + 0.993566i \(0.536127\pi\)
\(240\) 3.23797 0.209010
\(241\) −1.58795 −0.102289 −0.0511444 0.998691i \(-0.516287\pi\)
−0.0511444 + 0.998691i \(0.516287\pi\)
\(242\) −2.79650 −0.179766
\(243\) 18.2491 1.17068
\(244\) −3.55691 −0.227708
\(245\) 0 0
\(246\) 6.87930 0.438608
\(247\) −1.47068 −0.0935773
\(248\) −12.4983 −0.793642
\(249\) −36.7734 −2.33042
\(250\) 2.42160 0.153156
\(251\) 4.92676 0.310974 0.155487 0.987838i \(-0.450305\pi\)
0.155487 + 0.987838i \(0.450305\pi\)
\(252\) 0 0
\(253\) −13.1284 −0.825378
\(254\) 0.387890 0.0243384
\(255\) 1.55691 0.0974978
\(256\) 1.07162 0.0669764
\(257\) 8.01461 0.499938 0.249969 0.968254i \(-0.419580\pi\)
0.249969 + 0.968254i \(0.419580\pi\)
\(258\) −12.0621 −0.750952
\(259\) 0 0
\(260\) −0.941367 −0.0583811
\(261\) 10.7492 0.665361
\(262\) −4.99656 −0.308689
\(263\) 1.60256 0.0988179 0.0494090 0.998779i \(-0.484266\pi\)
0.0494090 + 0.998779i \(0.484266\pi\)
\(264\) −8.99656 −0.553700
\(265\) −5.96896 −0.366671
\(266\) 0 0
\(267\) 15.5423 0.951174
\(268\) 28.3449 1.73144
\(269\) 11.8207 0.720719 0.360359 0.932814i \(-0.382654\pi\)
0.360359 + 0.932814i \(0.382654\pi\)
\(270\) −0.527497 −0.0321024
\(271\) −21.8827 −1.32928 −0.664641 0.747163i \(-0.731415\pi\)
−0.664641 + 0.747163i \(0.731415\pi\)
\(272\) 3.55691 0.215670
\(273\) 0 0
\(274\) 5.34836 0.323106
\(275\) 10.6155 0.640142
\(276\) 23.3484 1.40541
\(277\) −10.2181 −0.613946 −0.306973 0.951718i \(-0.599316\pi\)
−0.306973 + 0.951718i \(0.599316\pi\)
\(278\) 6.55348 0.393051
\(279\) 14.4672 0.866131
\(280\) 0 0
\(281\) 1.54231 0.0920063 0.0460031 0.998941i \(-0.485352\pi\)
0.0460031 + 0.998941i \(0.485352\pi\)
\(282\) 9.08785 0.541174
\(283\) −15.8466 −0.941985 −0.470993 0.882137i \(-0.656104\pi\)
−0.470993 + 0.882137i \(0.656104\pi\)
\(284\) −2.11727 −0.125637
\(285\) 1.75086 0.103712
\(286\) 1.05863 0.0625983
\(287\) 0 0
\(288\) 9.95779 0.586769
\(289\) −15.2897 −0.899396
\(290\) −1.30090 −0.0763914
\(291\) −7.80605 −0.457599
\(292\) 13.5930 0.795470
\(293\) −11.0828 −0.647464 −0.323732 0.946149i \(-0.604938\pi\)
−0.323732 + 0.946149i \(0.604938\pi\)
\(294\) 0 0
\(295\) −6.44309 −0.375131
\(296\) 4.20855 0.244617
\(297\) −4.76203 −0.276321
\(298\) 4.38101 0.253785
\(299\) −5.83709 −0.337568
\(300\) −18.8793 −1.09000
\(301\) 0 0
\(302\) 3.32926 0.191577
\(303\) 17.4328 1.00149
\(304\) 4.00000 0.229416
\(305\) −1.05863 −0.0606172
\(306\) 1.26719 0.0724402
\(307\) −20.4121 −1.16498 −0.582489 0.812839i \(-0.697921\pi\)
−0.582489 + 0.812839i \(0.697921\pi\)
\(308\) 0 0
\(309\) −38.2277 −2.17470
\(310\) −1.75086 −0.0994421
\(311\) 1.92332 0.109062 0.0545308 0.998512i \(-0.482634\pi\)
0.0545308 + 0.998512i \(0.482634\pi\)
\(312\) −4.00000 −0.226455
\(313\) 14.3664 0.812037 0.406019 0.913865i \(-0.366917\pi\)
0.406019 + 0.913865i \(0.366917\pi\)
\(314\) 2.84320 0.160451
\(315\) 0 0
\(316\) 2.38101 0.133943
\(317\) 15.5569 0.873763 0.436882 0.899519i \(-0.356083\pi\)
0.436882 + 0.899519i \(0.356083\pi\)
\(318\) −11.9379 −0.669446
\(319\) −11.7440 −0.657537
\(320\) 1.67418 0.0935895
\(321\) 12.4983 0.697586
\(322\) 0 0
\(323\) 1.92332 0.107016
\(324\) 19.4526 1.08070
\(325\) 4.71982 0.261809
\(326\) 3.00344 0.166345
\(327\) −17.8207 −0.985485
\(328\) 11.5569 0.638124
\(329\) 0 0
\(330\) −1.26031 −0.0693778
\(331\) 31.5500 1.73415 0.867073 0.498180i \(-0.165998\pi\)
0.867073 + 0.498180i \(0.165998\pi\)
\(332\) −29.0777 −1.59585
\(333\) −4.87156 −0.266960
\(334\) 7.80605 0.427128
\(335\) 8.43621 0.460919
\(336\) 0 0
\(337\) −8.42666 −0.459029 −0.229515 0.973305i \(-0.573714\pi\)
−0.229515 + 0.973305i \(0.573714\pi\)
\(338\) 0.470683 0.0256018
\(339\) 22.2491 1.20841
\(340\) 1.23109 0.0667655
\(341\) −15.8061 −0.855946
\(342\) 1.42504 0.0770573
\(343\) 0 0
\(344\) −20.2637 −1.09255
\(345\) 6.94910 0.374127
\(346\) 10.9673 0.589608
\(347\) 19.3484 1.03867 0.519337 0.854569i \(-0.326179\pi\)
0.519337 + 0.854569i \(0.326179\pi\)
\(348\) 20.8862 1.11962
\(349\) −27.2553 −1.45894 −0.729470 0.684013i \(-0.760233\pi\)
−0.729470 + 0.684013i \(0.760233\pi\)
\(350\) 0 0
\(351\) −2.11727 −0.113011
\(352\) −10.8793 −0.579868
\(353\) 25.6742 1.36650 0.683249 0.730185i \(-0.260566\pi\)
0.683249 + 0.730185i \(0.260566\pi\)
\(354\) −12.8862 −0.684892
\(355\) −0.630155 −0.0334452
\(356\) 12.2897 0.651354
\(357\) 0 0
\(358\) 9.90422 0.523454
\(359\) −23.4182 −1.23596 −0.617982 0.786192i \(-0.712049\pi\)
−0.617982 + 0.786192i \(0.712049\pi\)
\(360\) 1.93793 0.102138
\(361\) −16.8371 −0.886163
\(362\) −7.88273 −0.414307
\(363\) 13.3630 0.701374
\(364\) 0 0
\(365\) 4.04564 0.211759
\(366\) −2.11727 −0.110671
\(367\) 14.6854 0.766569 0.383285 0.923630i \(-0.374793\pi\)
0.383285 + 0.923630i \(0.374793\pi\)
\(368\) 15.8759 0.827586
\(369\) −13.3776 −0.696409
\(370\) 0.589568 0.0306502
\(371\) 0 0
\(372\) 28.1104 1.45746
\(373\) −23.6673 −1.22545 −0.612723 0.790298i \(-0.709926\pi\)
−0.612723 + 0.790298i \(0.709926\pi\)
\(374\) −1.38445 −0.0715883
\(375\) −11.5715 −0.597551
\(376\) 15.2672 0.787345
\(377\) −5.22154 −0.268923
\(378\) 0 0
\(379\) −32.7405 −1.68177 −0.840884 0.541215i \(-0.817965\pi\)
−0.840884 + 0.541215i \(0.817965\pi\)
\(380\) 1.38445 0.0710209
\(381\) −1.85352 −0.0949586
\(382\) −3.50172 −0.179164
\(383\) 22.6155 1.15560 0.577800 0.816178i \(-0.303911\pi\)
0.577800 + 0.816178i \(0.303911\pi\)
\(384\) 25.1070 1.28123
\(385\) 0 0
\(386\) −0.706834 −0.0359769
\(387\) 23.4561 1.19234
\(388\) −6.17246 −0.313359
\(389\) 38.0483 1.92913 0.964563 0.263852i \(-0.0849930\pi\)
0.964563 + 0.263852i \(0.0849930\pi\)
\(390\) −0.560352 −0.0283745
\(391\) 7.63359 0.386047
\(392\) 0 0
\(393\) 23.8759 1.20438
\(394\) 11.2932 0.568941
\(395\) 0.708654 0.0356562
\(396\) 8.23453 0.413801
\(397\) 11.7052 0.587468 0.293734 0.955887i \(-0.405102\pi\)
0.293734 + 0.955887i \(0.405102\pi\)
\(398\) 0.948243 0.0475311
\(399\) 0 0
\(400\) −12.8371 −0.641855
\(401\) 3.55691 0.177624 0.0888119 0.996048i \(-0.471693\pi\)
0.0888119 + 0.996048i \(0.471693\pi\)
\(402\) 16.8724 0.841520
\(403\) −7.02760 −0.350070
\(404\) 13.7846 0.685808
\(405\) 5.78963 0.287689
\(406\) 0 0
\(407\) 5.32238 0.263821
\(408\) 5.23109 0.258978
\(409\) −5.26213 −0.260196 −0.130098 0.991501i \(-0.541529\pi\)
−0.130098 + 0.991501i \(0.541529\pi\)
\(410\) 1.61899 0.0799560
\(411\) −25.5569 −1.26063
\(412\) −30.2277 −1.48921
\(413\) 0 0
\(414\) 5.65593 0.277974
\(415\) −8.65432 −0.424824
\(416\) −4.83709 −0.237158
\(417\) −31.3155 −1.53353
\(418\) −1.55691 −0.0761512
\(419\) −26.0337 −1.27183 −0.635915 0.771759i \(-0.719377\pi\)
−0.635915 + 0.771759i \(0.719377\pi\)
\(420\) 0 0
\(421\) 22.2423 1.08402 0.542011 0.840372i \(-0.317663\pi\)
0.542011 + 0.840372i \(0.317663\pi\)
\(422\) 4.75430 0.231436
\(423\) −17.6724 −0.859260
\(424\) −20.0552 −0.973966
\(425\) −6.17246 −0.299408
\(426\) −1.26031 −0.0610622
\(427\) 0 0
\(428\) 9.88273 0.477700
\(429\) −5.05863 −0.244233
\(430\) −2.83871 −0.136895
\(431\) 27.6742 1.33302 0.666509 0.745497i \(-0.267788\pi\)
0.666509 + 0.745497i \(0.267788\pi\)
\(432\) 5.75859 0.277060
\(433\) 12.7880 0.614552 0.307276 0.951620i \(-0.400582\pi\)
0.307276 + 0.951620i \(0.400582\pi\)
\(434\) 0 0
\(435\) 6.21629 0.298048
\(436\) −14.0913 −0.674850
\(437\) 8.58451 0.410653
\(438\) 8.09129 0.386617
\(439\) 18.1656 0.866996 0.433498 0.901155i \(-0.357279\pi\)
0.433498 + 0.901155i \(0.357279\pi\)
\(440\) −2.11727 −0.100937
\(441\) 0 0
\(442\) −0.615547 −0.0292786
\(443\) 0.107714 0.00511767 0.00255883 0.999997i \(-0.499185\pi\)
0.00255883 + 0.999997i \(0.499185\pi\)
\(444\) −9.46563 −0.449219
\(445\) 3.65775 0.173394
\(446\) −4.77340 −0.226027
\(447\) −20.9345 −0.990167
\(448\) 0 0
\(449\) −22.1725 −1.04638 −0.523192 0.852215i \(-0.675259\pi\)
−0.523192 + 0.852215i \(0.675259\pi\)
\(450\) −4.57334 −0.215589
\(451\) 14.6155 0.688219
\(452\) 17.5930 0.827505
\(453\) −15.9087 −0.747457
\(454\) 2.53437 0.118944
\(455\) 0 0
\(456\) 5.88273 0.275484
\(457\) 4.35953 0.203930 0.101965 0.994788i \(-0.467487\pi\)
0.101965 + 0.994788i \(0.467487\pi\)
\(458\) −1.56379 −0.0730711
\(459\) 2.76891 0.129241
\(460\) 5.49484 0.256198
\(461\) −32.3810 −1.50813 −0.754067 0.656797i \(-0.771911\pi\)
−0.754067 + 0.656797i \(0.771911\pi\)
\(462\) 0 0
\(463\) 8.36641 0.388820 0.194410 0.980920i \(-0.437721\pi\)
0.194410 + 0.980920i \(0.437721\pi\)
\(464\) 14.2017 0.659296
\(465\) 8.36641 0.387983
\(466\) 6.45769 0.299147
\(467\) 19.5423 0.904310 0.452155 0.891939i \(-0.350655\pi\)
0.452155 + 0.891939i \(0.350655\pi\)
\(468\) 3.66119 0.169239
\(469\) 0 0
\(470\) 2.13875 0.0986532
\(471\) −13.5861 −0.626016
\(472\) −21.6482 −0.996439
\(473\) −25.6267 −1.17832
\(474\) 1.41731 0.0650991
\(475\) −6.94137 −0.318492
\(476\) 0 0
\(477\) 23.2147 1.06293
\(478\) −1.64820 −0.0753870
\(479\) 28.5224 1.30322 0.651612 0.758553i \(-0.274093\pi\)
0.651612 + 0.758553i \(0.274093\pi\)
\(480\) 5.75859 0.262843
\(481\) 2.36641 0.107899
\(482\) −0.747422 −0.0340441
\(483\) 0 0
\(484\) 10.5665 0.480294
\(485\) −1.83709 −0.0834180
\(486\) 8.58957 0.389631
\(487\) 24.8241 1.12489 0.562444 0.826836i \(-0.309861\pi\)
0.562444 + 0.826836i \(0.309861\pi\)
\(488\) −3.55691 −0.161014
\(489\) −14.3518 −0.649011
\(490\) 0 0
\(491\) 29.1690 1.31638 0.658190 0.752852i \(-0.271322\pi\)
0.658190 + 0.752852i \(0.271322\pi\)
\(492\) −25.9931 −1.17186
\(493\) 6.82860 0.307545
\(494\) −0.692226 −0.0311447
\(495\) 2.45082 0.110156
\(496\) 19.1138 0.858236
\(497\) 0 0
\(498\) −17.3086 −0.775618
\(499\) −33.3009 −1.49075 −0.745376 0.666644i \(-0.767730\pi\)
−0.745376 + 0.666644i \(0.767730\pi\)
\(500\) −9.14992 −0.409197
\(501\) −37.3009 −1.66648
\(502\) 2.31894 0.103500
\(503\) 12.3258 0.549581 0.274791 0.961504i \(-0.411391\pi\)
0.274791 + 0.961504i \(0.411391\pi\)
\(504\) 0 0
\(505\) 4.10266 0.182566
\(506\) −6.17934 −0.274705
\(507\) −2.24914 −0.0998878
\(508\) −1.46563 −0.0650267
\(509\) −23.7052 −1.05072 −0.525358 0.850882i \(-0.676068\pi\)
−0.525358 + 0.850882i \(0.676068\pi\)
\(510\) 0.732814 0.0324495
\(511\) 0 0
\(512\) 22.8302 1.00896
\(513\) 3.11383 0.137479
\(514\) 3.77234 0.166391
\(515\) −8.99656 −0.396436
\(516\) 45.5760 2.00637
\(517\) 19.3078 0.849155
\(518\) 0 0
\(519\) −52.4070 −2.30041
\(520\) −0.941367 −0.0412817
\(521\) −43.9018 −1.92337 −0.961687 0.274149i \(-0.911604\pi\)
−0.961687 + 0.274149i \(0.911604\pi\)
\(522\) 5.05949 0.221448
\(523\) −37.4328 −1.63682 −0.818410 0.574634i \(-0.805144\pi\)
−0.818410 + 0.574634i \(0.805144\pi\)
\(524\) 18.8793 0.824746
\(525\) 0 0
\(526\) 0.754297 0.0328889
\(527\) 9.19051 0.400345
\(528\) 13.7586 0.598766
\(529\) 11.0716 0.481375
\(530\) −2.80949 −0.122037
\(531\) 25.0586 1.08745
\(532\) 0 0
\(533\) 6.49828 0.281472
\(534\) 7.31551 0.316573
\(535\) 2.94137 0.127166
\(536\) 28.3449 1.22431
\(537\) −47.3269 −2.04231
\(538\) 5.56379 0.239872
\(539\) 0 0
\(540\) 1.99312 0.0857704
\(541\) −34.9751 −1.50370 −0.751848 0.659336i \(-0.770837\pi\)
−0.751848 + 0.659336i \(0.770837\pi\)
\(542\) −10.2998 −0.442416
\(543\) 37.6673 1.61646
\(544\) 6.32582 0.271217
\(545\) −4.19395 −0.179649
\(546\) 0 0
\(547\) 6.50783 0.278255 0.139127 0.990274i \(-0.455570\pi\)
0.139127 + 0.990274i \(0.455570\pi\)
\(548\) −20.2086 −0.863267
\(549\) 4.11727 0.175721
\(550\) 4.99656 0.213054
\(551\) 7.67924 0.327146
\(552\) 23.3484 0.993772
\(553\) 0 0
\(554\) −4.80949 −0.204336
\(555\) −2.81722 −0.119585
\(556\) −24.7620 −1.05014
\(557\) −43.4328 −1.84031 −0.920153 0.391559i \(-0.871936\pi\)
−0.920153 + 0.391559i \(0.871936\pi\)
\(558\) 6.80949 0.288269
\(559\) −11.3940 −0.481915
\(560\) 0 0
\(561\) 6.61555 0.279309
\(562\) 0.725938 0.0306218
\(563\) 33.8827 1.42799 0.713993 0.700152i \(-0.246885\pi\)
0.713993 + 0.700152i \(0.246885\pi\)
\(564\) −34.3380 −1.44589
\(565\) 5.23615 0.220287
\(566\) −7.45875 −0.313515
\(567\) 0 0
\(568\) −2.11727 −0.0888385
\(569\) 19.2147 0.805521 0.402760 0.915305i \(-0.368051\pi\)
0.402760 + 0.915305i \(0.368051\pi\)
\(570\) 0.824101 0.0345178
\(571\) 20.8268 0.871573 0.435787 0.900050i \(-0.356470\pi\)
0.435787 + 0.900050i \(0.356470\pi\)
\(572\) −4.00000 −0.167248
\(573\) 16.7328 0.699023
\(574\) 0 0
\(575\) −27.5500 −1.14892
\(576\) −6.51127 −0.271303
\(577\) −28.6448 −1.19250 −0.596249 0.802800i \(-0.703343\pi\)
−0.596249 + 0.802800i \(0.703343\pi\)
\(578\) −7.19662 −0.299340
\(579\) 3.37758 0.140367
\(580\) 4.91539 0.204100
\(581\) 0 0
\(582\) −3.67418 −0.152300
\(583\) −25.3630 −1.05043
\(584\) 13.5930 0.562483
\(585\) 1.08967 0.0450523
\(586\) −5.21649 −0.215491
\(587\) 4.32076 0.178337 0.0891685 0.996017i \(-0.471579\pi\)
0.0891685 + 0.996017i \(0.471579\pi\)
\(588\) 0 0
\(589\) 10.3354 0.425862
\(590\) −3.03265 −0.124852
\(591\) −53.9639 −2.21978
\(592\) −6.43621 −0.264527
\(593\) 15.9690 0.655767 0.327883 0.944718i \(-0.393665\pi\)
0.327883 + 0.944718i \(0.393665\pi\)
\(594\) −2.24141 −0.0919661
\(595\) 0 0
\(596\) −16.5535 −0.678057
\(597\) −4.53114 −0.185447
\(598\) −2.74742 −0.112350
\(599\) −16.8697 −0.689279 −0.344640 0.938735i \(-0.611999\pi\)
−0.344640 + 0.938735i \(0.611999\pi\)
\(600\) −18.8793 −0.770744
\(601\) 15.3415 0.625792 0.312896 0.949787i \(-0.398701\pi\)
0.312896 + 0.949787i \(0.398701\pi\)
\(602\) 0 0
\(603\) −32.8103 −1.33614
\(604\) −12.5795 −0.511851
\(605\) 3.14486 0.127857
\(606\) 8.20532 0.333318
\(607\) −35.8353 −1.45451 −0.727254 0.686368i \(-0.759204\pi\)
−0.727254 + 0.686368i \(0.759204\pi\)
\(608\) 7.11383 0.288504
\(609\) 0 0
\(610\) −0.498281 −0.0201748
\(611\) 8.58451 0.347292
\(612\) −4.78801 −0.193544
\(613\) 19.6673 0.794355 0.397177 0.917742i \(-0.369990\pi\)
0.397177 + 0.917742i \(0.369990\pi\)
\(614\) −9.60761 −0.387732
\(615\) −7.73625 −0.311956
\(616\) 0 0
\(617\) −41.4588 −1.66907 −0.834533 0.550958i \(-0.814263\pi\)
−0.834533 + 0.550958i \(0.814263\pi\)
\(618\) −17.9931 −0.723790
\(619\) −10.8793 −0.437276 −0.218638 0.975806i \(-0.570161\pi\)
−0.218638 + 0.975806i \(0.570161\pi\)
\(620\) 6.61555 0.265687
\(621\) 12.3587 0.495937
\(622\) 0.905275 0.0362982
\(623\) 0 0
\(624\) 6.11727 0.244887
\(625\) 20.8759 0.835034
\(626\) 6.76203 0.270265
\(627\) 7.43965 0.297111
\(628\) −10.7429 −0.428689
\(629\) −3.09472 −0.123395
\(630\) 0 0
\(631\) −31.4396 −1.25159 −0.625796 0.779987i \(-0.715226\pi\)
−0.625796 + 0.779987i \(0.715226\pi\)
\(632\) 2.38101 0.0947117
\(633\) −22.7182 −0.902968
\(634\) 7.32238 0.290809
\(635\) −0.436210 −0.0173105
\(636\) 45.1070 1.78861
\(637\) 0 0
\(638\) −5.52770 −0.218844
\(639\) 2.45082 0.0969529
\(640\) 5.90871 0.233562
\(641\) 3.04221 0.120160 0.0600799 0.998194i \(-0.480864\pi\)
0.0600799 + 0.998194i \(0.480864\pi\)
\(642\) 5.88273 0.232173
\(643\) −8.02922 −0.316641 −0.158321 0.987388i \(-0.550608\pi\)
−0.158321 + 0.987388i \(0.550608\pi\)
\(644\) 0 0
\(645\) 13.5646 0.534107
\(646\) 0.905275 0.0356176
\(647\) −7.07324 −0.278078 −0.139039 0.990287i \(-0.544401\pi\)
−0.139039 + 0.990287i \(0.544401\pi\)
\(648\) 19.4526 0.764172
\(649\) −27.3776 −1.07466
\(650\) 2.22154 0.0871361
\(651\) 0 0
\(652\) −11.3484 −0.444436
\(653\) −15.7586 −0.616681 −0.308341 0.951276i \(-0.599774\pi\)
−0.308341 + 0.951276i \(0.599774\pi\)
\(654\) −8.38789 −0.327992
\(655\) 5.61899 0.219552
\(656\) −17.6742 −0.690061
\(657\) −15.7344 −0.613859
\(658\) 0 0
\(659\) −12.2181 −0.475950 −0.237975 0.971271i \(-0.576484\pi\)
−0.237975 + 0.971271i \(0.576484\pi\)
\(660\) 4.76203 0.185362
\(661\) −3.73443 −0.145253 −0.0726263 0.997359i \(-0.523138\pi\)
−0.0726263 + 0.997359i \(0.523138\pi\)
\(662\) 14.8501 0.577165
\(663\) 2.94137 0.114233
\(664\) −29.0777 −1.12844
\(665\) 0 0
\(666\) −2.29296 −0.0888506
\(667\) 30.4786 1.18014
\(668\) −29.4948 −1.14119
\(669\) 22.8095 0.881866
\(670\) 3.97078 0.153405
\(671\) −4.49828 −0.173654
\(672\) 0 0
\(673\) −5.65775 −0.218090 −0.109045 0.994037i \(-0.534779\pi\)
−0.109045 + 0.994037i \(0.534779\pi\)
\(674\) −3.96629 −0.152776
\(675\) −9.99312 −0.384636
\(676\) −1.77846 −0.0684022
\(677\) −9.39906 −0.361235 −0.180618 0.983553i \(-0.557810\pi\)
−0.180618 + 0.983553i \(0.557810\pi\)
\(678\) 10.4723 0.402186
\(679\) 0 0
\(680\) 1.23109 0.0472103
\(681\) −12.1104 −0.464071
\(682\) −7.43965 −0.284879
\(683\) −20.7328 −0.793319 −0.396660 0.917966i \(-0.629831\pi\)
−0.396660 + 0.917966i \(0.629831\pi\)
\(684\) −5.38445 −0.205880
\(685\) −6.01461 −0.229806
\(686\) 0 0
\(687\) 7.47250 0.285094
\(688\) 30.9897 1.18147
\(689\) −11.2767 −0.429610
\(690\) 3.27083 0.124518
\(691\) −16.0862 −0.611949 −0.305975 0.952040i \(-0.598982\pi\)
−0.305975 + 0.952040i \(0.598982\pi\)
\(692\) −41.4396 −1.57530
\(693\) 0 0
\(694\) 9.10695 0.345695
\(695\) −7.36984 −0.279554
\(696\) 20.8862 0.791688
\(697\) −8.49828 −0.321895
\(698\) −12.8286 −0.485570
\(699\) −30.8578 −1.16715
\(700\) 0 0
\(701\) −6.98013 −0.263636 −0.131818 0.991274i \(-0.542081\pi\)
−0.131818 + 0.991274i \(0.542081\pi\)
\(702\) −0.996562 −0.0376128
\(703\) −3.48024 −0.131260
\(704\) 7.11383 0.268112
\(705\) −10.2199 −0.384905
\(706\) 12.0844 0.454803
\(707\) 0 0
\(708\) 48.6898 1.82988
\(709\) 8.39239 0.315183 0.157591 0.987504i \(-0.449627\pi\)
0.157591 + 0.987504i \(0.449627\pi\)
\(710\) −0.296604 −0.0111313
\(711\) −2.75612 −0.103362
\(712\) 12.2897 0.460577
\(713\) 41.0207 1.53624
\(714\) 0 0
\(715\) −1.19051 −0.0445225
\(716\) −37.4227 −1.39855
\(717\) 7.87586 0.294129
\(718\) −11.0225 −0.411358
\(719\) 5.16129 0.192484 0.0962418 0.995358i \(-0.469318\pi\)
0.0962418 + 0.995358i \(0.469318\pi\)
\(720\) −2.96371 −0.110451
\(721\) 0 0
\(722\) −7.92494 −0.294936
\(723\) 3.57152 0.132826
\(724\) 29.7846 1.10693
\(725\) −24.6448 −0.915284
\(726\) 6.28973 0.233434
\(727\) 40.4362 1.49970 0.749848 0.661610i \(-0.230127\pi\)
0.749848 + 0.661610i \(0.230127\pi\)
\(728\) 0 0
\(729\) −8.23109 −0.304855
\(730\) 1.90422 0.0704782
\(731\) 14.9008 0.551125
\(732\) 8.00000 0.295689
\(733\) 39.1311 1.44534 0.722670 0.691193i \(-0.242915\pi\)
0.722670 + 0.691193i \(0.242915\pi\)
\(734\) 6.91215 0.255132
\(735\) 0 0
\(736\) 28.2345 1.04074
\(737\) 35.8466 1.32043
\(738\) −6.29660 −0.231781
\(739\) −7.13531 −0.262477 −0.131238 0.991351i \(-0.541895\pi\)
−0.131238 + 0.991351i \(0.541895\pi\)
\(740\) −2.22766 −0.0818903
\(741\) 3.30777 0.121514
\(742\) 0 0
\(743\) 13.8827 0.509308 0.254654 0.967032i \(-0.418038\pi\)
0.254654 + 0.967032i \(0.418038\pi\)
\(744\) 28.1104 1.03058
\(745\) −4.92676 −0.180502
\(746\) −11.1398 −0.407857
\(747\) 33.6586 1.23150
\(748\) 5.23109 0.191268
\(749\) 0 0
\(750\) −5.44652 −0.198879
\(751\) −37.3251 −1.36201 −0.681005 0.732278i \(-0.738457\pi\)
−0.681005 + 0.732278i \(0.738457\pi\)
\(752\) −23.3484 −0.851427
\(753\) −11.0810 −0.403813
\(754\) −2.45769 −0.0895039
\(755\) −3.74398 −0.136258
\(756\) 0 0
\(757\) −7.10428 −0.258209 −0.129105 0.991631i \(-0.541210\pi\)
−0.129105 + 0.991631i \(0.541210\pi\)
\(758\) −15.4104 −0.559732
\(759\) 29.5277 1.07179
\(760\) 1.38445 0.0502194
\(761\) 25.9621 0.941125 0.470562 0.882367i \(-0.344051\pi\)
0.470562 + 0.882367i \(0.344051\pi\)
\(762\) −0.872420 −0.0316044
\(763\) 0 0
\(764\) 13.2311 0.478684
\(765\) −1.42504 −0.0515224
\(766\) 10.6448 0.384611
\(767\) −12.1725 −0.439522
\(768\) −2.41023 −0.0869717
\(769\) 21.4638 0.774005 0.387002 0.922079i \(-0.373511\pi\)
0.387002 + 0.922079i \(0.373511\pi\)
\(770\) 0 0
\(771\) −18.0260 −0.649190
\(772\) 2.67074 0.0961221
\(773\) 40.4914 1.45637 0.728187 0.685378i \(-0.240363\pi\)
0.728187 + 0.685378i \(0.240363\pi\)
\(774\) 11.0404 0.396838
\(775\) −33.1690 −1.19147
\(776\) −6.17246 −0.221578
\(777\) 0 0
\(778\) 17.9087 0.642058
\(779\) −9.55691 −0.342412
\(780\) 2.11727 0.0758103
\(781\) −2.67762 −0.0958127
\(782\) 3.59301 0.128486
\(783\) 11.0554 0.395088
\(784\) 0 0
\(785\) −3.19738 −0.114119
\(786\) 11.2380 0.400845
\(787\) 5.02072 0.178969 0.0894847 0.995988i \(-0.471478\pi\)
0.0894847 + 0.995988i \(0.471478\pi\)
\(788\) −42.6707 −1.52008
\(789\) −3.60438 −0.128319
\(790\) 0.333552 0.0118672
\(791\) 0 0
\(792\) 8.23453 0.292601
\(793\) −2.00000 −0.0710221
\(794\) 5.50945 0.195523
\(795\) 13.4250 0.476137
\(796\) −3.58289 −0.126992
\(797\) −19.7002 −0.697815 −0.348908 0.937157i \(-0.613447\pi\)
−0.348908 + 0.937157i \(0.613447\pi\)
\(798\) 0 0
\(799\) −11.2266 −0.397169
\(800\) −22.8302 −0.807170
\(801\) −14.2258 −0.502645
\(802\) 1.67418 0.0591174
\(803\) 17.1905 0.606640
\(804\) −63.7517 −2.24835
\(805\) 0 0
\(806\) −3.30777 −0.116511
\(807\) −26.5863 −0.935883
\(808\) 13.7846 0.484940
\(809\) −2.57678 −0.0905947 −0.0452974 0.998974i \(-0.514424\pi\)
−0.0452974 + 0.998974i \(0.514424\pi\)
\(810\) 2.72508 0.0957496
\(811\) −41.2311 −1.44782 −0.723910 0.689895i \(-0.757657\pi\)
−0.723910 + 0.689895i \(0.757657\pi\)
\(812\) 0 0
\(813\) 49.2173 1.72613
\(814\) 2.50516 0.0878057
\(815\) −3.37758 −0.118311
\(816\) −8.00000 −0.280056
\(817\) 16.7570 0.586252
\(818\) −2.47680 −0.0865992
\(819\) 0 0
\(820\) −6.11727 −0.213624
\(821\) 5.26719 0.183826 0.0919130 0.995767i \(-0.470702\pi\)
0.0919130 + 0.995767i \(0.470702\pi\)
\(822\) −12.0292 −0.419567
\(823\) 36.0191 1.25555 0.627774 0.778396i \(-0.283966\pi\)
0.627774 + 0.778396i \(0.283966\pi\)
\(824\) −30.2277 −1.05303
\(825\) −23.8759 −0.831251
\(826\) 0 0
\(827\) −16.3157 −0.567353 −0.283676 0.958920i \(-0.591554\pi\)
−0.283676 + 0.958920i \(0.591554\pi\)
\(828\) −21.3707 −0.742683
\(829\) 30.3956 1.05568 0.527842 0.849343i \(-0.323001\pi\)
0.527842 + 0.849343i \(0.323001\pi\)
\(830\) −4.07344 −0.141391
\(831\) 22.9820 0.797235
\(832\) 3.16291 0.109654
\(833\) 0 0
\(834\) −14.7397 −0.510394
\(835\) −8.77846 −0.303791
\(836\) 5.88273 0.203459
\(837\) 14.8793 0.514304
\(838\) −12.2536 −0.423295
\(839\) −29.8398 −1.03018 −0.515092 0.857135i \(-0.672242\pi\)
−0.515092 + 0.857135i \(0.672242\pi\)
\(840\) 0 0
\(841\) −1.73549 −0.0598445
\(842\) 10.4691 0.360788
\(843\) −3.46886 −0.119474
\(844\) −17.9639 −0.618343
\(845\) −0.529317 −0.0182090
\(846\) −8.31809 −0.285982
\(847\) 0 0
\(848\) 30.6707 1.05324
\(849\) 35.6413 1.22321
\(850\) −2.90528 −0.0996501
\(851\) −13.8129 −0.473501
\(852\) 4.76203 0.163144
\(853\) 0.203497 0.00696761 0.00348380 0.999994i \(-0.498891\pi\)
0.00348380 + 0.999994i \(0.498891\pi\)
\(854\) 0 0
\(855\) −1.60256 −0.0548063
\(856\) 9.88273 0.337785
\(857\) 12.6155 0.430939 0.215469 0.976511i \(-0.430872\pi\)
0.215469 + 0.976511i \(0.430872\pi\)
\(858\) −2.38101 −0.0812865
\(859\) 27.9671 0.954227 0.477113 0.878842i \(-0.341683\pi\)
0.477113 + 0.878842i \(0.341683\pi\)
\(860\) 10.7259 0.365751
\(861\) 0 0
\(862\) 13.0258 0.443660
\(863\) −2.76891 −0.0942546 −0.0471273 0.998889i \(-0.515007\pi\)
−0.0471273 + 0.998889i \(0.515007\pi\)
\(864\) 10.2414 0.348420
\(865\) −12.3336 −0.419353
\(866\) 6.01910 0.204537
\(867\) 34.3887 1.16790
\(868\) 0 0
\(869\) 3.01117 0.102147
\(870\) 2.92590 0.0991974
\(871\) 15.9379 0.540036
\(872\) −14.0913 −0.477191
\(873\) 7.14486 0.241817
\(874\) 4.04059 0.136675
\(875\) 0 0
\(876\) −30.5726 −1.03295
\(877\) 6.71133 0.226626 0.113313 0.993559i \(-0.463854\pi\)
0.113313 + 0.993559i \(0.463854\pi\)
\(878\) 8.55024 0.288557
\(879\) 24.9268 0.840759
\(880\) 3.23797 0.109152
\(881\) 18.3741 0.619040 0.309520 0.950893i \(-0.399832\pi\)
0.309520 + 0.950893i \(0.399832\pi\)
\(882\) 0 0
\(883\) 9.93105 0.334207 0.167103 0.985939i \(-0.446559\pi\)
0.167103 + 0.985939i \(0.446559\pi\)
\(884\) 2.32582 0.0782258
\(885\) 14.4914 0.487123
\(886\) 0.0506994 0.00170328
\(887\) 51.3776 1.72509 0.862545 0.505980i \(-0.168869\pi\)
0.862545 + 0.505980i \(0.168869\pi\)
\(888\) −9.46563 −0.317646
\(889\) 0 0
\(890\) 1.72164 0.0577096
\(891\) 24.6009 0.824162
\(892\) 18.0361 0.603893
\(893\) −12.6251 −0.422483
\(894\) −9.85352 −0.329551
\(895\) −11.1380 −0.372302
\(896\) 0 0
\(897\) 13.1284 0.438346
\(898\) −10.4362 −0.348261
\(899\) 36.6949 1.22384
\(900\) 17.2802 0.576006
\(901\) 14.7474 0.491308
\(902\) 6.87930 0.229055
\(903\) 0 0
\(904\) 17.5930 0.585135
\(905\) 8.86469 0.294672
\(906\) −7.48797 −0.248771
\(907\) 4.34225 0.144182 0.0720910 0.997398i \(-0.477033\pi\)
0.0720910 + 0.997398i \(0.477033\pi\)
\(908\) −9.57602 −0.317791
\(909\) −15.9562 −0.529233
\(910\) 0 0
\(911\) 31.4853 1.04315 0.521577 0.853204i \(-0.325344\pi\)
0.521577 + 0.853204i \(0.325344\pi\)
\(912\) −8.99656 −0.297906
\(913\) −36.7734 −1.21702
\(914\) 2.05196 0.0678728
\(915\) 2.38101 0.0787139
\(916\) 5.90871 0.195229
\(917\) 0 0
\(918\) 1.30328 0.0430146
\(919\) 43.4588 1.43357 0.716786 0.697293i \(-0.245612\pi\)
0.716786 + 0.697293i \(0.245612\pi\)
\(920\) 5.49484 0.181160
\(921\) 45.9096 1.51277
\(922\) −15.2412 −0.501942
\(923\) −1.19051 −0.0391860
\(924\) 0 0
\(925\) 11.1690 0.367235
\(926\) 3.93793 0.129408
\(927\) 34.9897 1.14921
\(928\) 25.2571 0.829104
\(929\) −4.40517 −0.144529 −0.0722645 0.997385i \(-0.523023\pi\)
−0.0722645 + 0.997385i \(0.523023\pi\)
\(930\) 3.93793 0.129130
\(931\) 0 0
\(932\) −24.4001 −0.799252
\(933\) −4.32582 −0.141621
\(934\) 9.19824 0.300976
\(935\) 1.55691 0.0509165
\(936\) 3.66119 0.119670
\(937\) 34.0990 1.11397 0.556983 0.830524i \(-0.311959\pi\)
0.556983 + 0.830524i \(0.311959\pi\)
\(938\) 0 0
\(939\) −32.3121 −1.05446
\(940\) −8.08117 −0.263579
\(941\) −44.4672 −1.44959 −0.724795 0.688964i \(-0.758066\pi\)
−0.724795 + 0.688964i \(0.758066\pi\)
\(942\) −6.39477 −0.208353
\(943\) −37.9311 −1.23521
\(944\) 33.1070 1.07754
\(945\) 0 0
\(946\) −12.0621 −0.392172
\(947\) 57.9311 1.88251 0.941253 0.337702i \(-0.109650\pi\)
0.941253 + 0.337702i \(0.109650\pi\)
\(948\) −5.35524 −0.173930
\(949\) 7.64315 0.248107
\(950\) −3.26719 −0.106002
\(951\) −34.9897 −1.13462
\(952\) 0 0
\(953\) 35.3060 1.14367 0.571836 0.820368i \(-0.306231\pi\)
0.571836 + 0.820368i \(0.306231\pi\)
\(954\) 10.9268 0.353767
\(955\) 3.93793 0.127428
\(956\) 6.22766 0.201417
\(957\) 26.4139 0.853839
\(958\) 13.4250 0.433743
\(959\) 0 0
\(960\) −3.76547 −0.121530
\(961\) 18.3871 0.593133
\(962\) 1.11383 0.0359113
\(963\) −11.4396 −0.368638
\(964\) 2.82410 0.0909582
\(965\) 0.794885 0.0255882
\(966\) 0 0
\(967\) 23.7148 0.762616 0.381308 0.924448i \(-0.375474\pi\)
0.381308 + 0.924448i \(0.375474\pi\)
\(968\) 10.5665 0.339619
\(969\) −4.32582 −0.138965
\(970\) −0.864688 −0.0277635
\(971\) 23.9379 0.768205 0.384102 0.923291i \(-0.374511\pi\)
0.384102 + 0.923291i \(0.374511\pi\)
\(972\) −32.4553 −1.04100
\(973\) 0 0
\(974\) 11.6843 0.374389
\(975\) −10.6155 −0.339970
\(976\) 5.43965 0.174119
\(977\) −16.1871 −0.517870 −0.258935 0.965895i \(-0.583372\pi\)
−0.258935 + 0.965895i \(0.583372\pi\)
\(978\) −6.75515 −0.216006
\(979\) 15.5423 0.496734
\(980\) 0 0
\(981\) 16.3112 0.520777
\(982\) 13.7294 0.438122
\(983\) −45.7243 −1.45838 −0.729190 0.684312i \(-0.760103\pi\)
−0.729190 + 0.684312i \(0.760103\pi\)
\(984\) −25.9931 −0.828631
\(985\) −12.7000 −0.404654
\(986\) 3.21411 0.102358
\(987\) 0 0
\(988\) 2.61555 0.0832116
\(989\) 66.5078 2.11483
\(990\) 1.15356 0.0366625
\(991\) −5.40356 −0.171650 −0.0858248 0.996310i \(-0.527353\pi\)
−0.0858248 + 0.996310i \(0.527353\pi\)
\(992\) 33.9931 1.07928
\(993\) −70.9605 −2.25186
\(994\) 0 0
\(995\) −1.06637 −0.0338061
\(996\) 65.3999 2.07228
\(997\) −6.04832 −0.191552 −0.0957761 0.995403i \(-0.530533\pi\)
−0.0957761 + 0.995403i \(0.530533\pi\)
\(998\) −15.6742 −0.496158
\(999\) −5.01031 −0.158519
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 637.2.a.j.1.2 3
3.2 odd 2 5733.2.a.x.1.2 3
7.2 even 3 637.2.e.i.508.2 6
7.3 odd 6 637.2.e.j.79.2 6
7.4 even 3 637.2.e.i.79.2 6
7.5 odd 6 637.2.e.j.508.2 6
7.6 odd 2 91.2.a.d.1.2 3
13.12 even 2 8281.2.a.bg.1.2 3
21.20 even 2 819.2.a.i.1.2 3
28.27 even 2 1456.2.a.t.1.1 3
35.34 odd 2 2275.2.a.m.1.2 3
56.13 odd 2 5824.2.a.by.1.1 3
56.27 even 2 5824.2.a.bs.1.3 3
91.34 even 4 1183.2.c.f.337.4 6
91.83 even 4 1183.2.c.f.337.3 6
91.90 odd 2 1183.2.a.i.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.2 3 7.6 odd 2
637.2.a.j.1.2 3 1.1 even 1 trivial
637.2.e.i.79.2 6 7.4 even 3
637.2.e.i.508.2 6 7.2 even 3
637.2.e.j.79.2 6 7.3 odd 6
637.2.e.j.508.2 6 7.5 odd 6
819.2.a.i.1.2 3 21.20 even 2
1183.2.a.i.1.2 3 91.90 odd 2
1183.2.c.f.337.3 6 91.83 even 4
1183.2.c.f.337.4 6 91.34 even 4
1456.2.a.t.1.1 3 28.27 even 2
2275.2.a.m.1.2 3 35.34 odd 2
5733.2.a.x.1.2 3 3.2 odd 2
5824.2.a.bs.1.3 3 56.27 even 2
5824.2.a.by.1.1 3 56.13 odd 2
8281.2.a.bg.1.2 3 13.12 even 2